Some Studies On Completely Regular Semigroups

The main topics are properties and structures of some sorts of completely regular semigroups and local descriptions of the lattice of varieties of completely regular semigroups.Firstly,based on two equivalent descriptions of translational hulls of completely simple semigroups,one of which is described in terms of Rees matrix semigroups with normalized sandwich matrices,and two given structure theorem of completely regular semigroups,another construction of completely regular semigroups,in which every(?)-class is a Rees matrix semigroup with norrealized sandwich matrix,is obtained.This construction is more concrete than the above two structure theorems.Moreover,a new construction of cryptogroups is presented.As applications,the proofs of two other structure theorems of cryptogroups are simplified.Secondly,using certain kinds of set-valued mappings,a refined semilattice of semigroups is equivalently rebuilt,and a briefer and more elegant proof of the associativity of the operation defined on the union of these semigroups is given.As applications,refined semilattices of semigroups are used to describe the structures of regular(respectively,right quasi-normal) orthocryp-togroups, Clifford semigroups,regular(respectively,right quasi-normal) bands and so on.Thirdly,applying a structure theorem of cryptogroups and congruence methods,the 18-element sublattice of the lattice of varieties of completely regular semigroups generated by the variety of normal orthocryptogroups,the variety of regular orthocryptogroups,the variety of orthocryptogroups,the variety of normal overabelian cryptogroups,the variety of regular overabelian cryptogroups and the variety of overabelian cryptogroups is determined.Certain counterexamples are given where necessary since some new varieties are produced.Lastly,Green's relation L on a band is not a congruence(i.e.not a left congruence) if and only if it contains either a certain 5-element subband or a certain 8-element subband. By proving this interesting result,a new fact on band semirings is obtained:the additive reduct of any band semiring is a regular band.